Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C26 |
| direct product | C2×C9×#419 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C2×D4 |
| commutator | C4 | C23×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C23×C3 | C2×C3×D4 |
| prime radical | C24×C3 | C3×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ C23×C4×C9 ◁ G | |
1128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2)(23,24,25,26,27,28,29,30,31);(3,4,5,6)(7,8)(9,16)(10,14)(11,20)(12,15)(13,18)(17,21)(19,22);(7,9,12,19)(8,13,15,21)(10,18,11,17)(14,22,20,16);(7,10,12,11)(8,14,15,20)(9,17,19,18)(13,16,21,22)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1626);